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2015, 2015(special): 1060-1069. doi: 10.3934/proc.2015.1060

## Simulation of complex dynamics using POD 'on the fly' and residual estimates

 1 Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain 2 E.T.S.I. Aeronáutica y del Espacio, Universidad Politécnica de Madrid, 28040 Madrid, Spain

Received  September 2014 Revised  February 2015 Published  November 2015

Proper orthogonal decomposition (POD) is a very effective means to identify dynamical information contained in sets of snapshots that cover numerically computed trajectories of dissipative systems of partial differential equations. Such information is organized in a hierarchy of POD modes and the system can be Galerkin-projected onto the associated linear subspace. Quite frequently, the outcome is a low dimensional model of the problem. Flexibility and efficiency of the approximation can be enhanced if POD is applied `on the fly', adaptively combining a standard numerical solver (which provides the necessary snapshots) with the reduced system in interspersed intervals, as both time and a bifurcation parameter are varied. Residual estimates are introduced to make this adaptation accurate and robust, preventing possible mode truncation instabilities in the presence of complex dynamics. All ideas are illustrated in some bifurcation scenarios including quasi-periodic and chaotic attractors, which highlights a good computational efficiency.
Citation: Filippo Terragni, José M. Vega. Simulation of complex dynamics using POD 'on the fly' and residual estimates. Conference Publications, 2015, 2015 (special) : 1060-1069. doi: 10.3934/proc.2015.1060
##### References:
 [1] E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, Classics Appl. Math. 45, SIAM, Philadelphia, 2003. [2] D. Alonso, J. M. Vega and A. Velazquez, Reduced order model for viscous aerodynamic flow past an airfoil, AIAA J., 48 (2010), 1946-1958. [3] D. Alonso, J. M. Vega, A. Velazquez and V. de Pablo, Reduced-order modeling of three-dimensional external aerodynamic flows, J. Aerospace Engrg., 25 (2012), 588-599. [4] I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 100-142. [5] P. Astrid, S. Weiland, K. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automatic Control, 53 (2008), 2237-2251. [6] M. J. Balajewicz, E. H. Dowell and B. R. Noack, Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation, J. Fluid Mech., 729 (2013), 285-308. [7] M. Bergmann, C. H. Bruneau and A. Iollo, Enablers for robust POD models, J. Comput. Phys., 228 (2009), 516-538. [8] T. Braconnier, M. Ferrier, J. C. Jouhaud, M. Montagnac and P. Sagaut, Towards an adaptive POD/SVD surrogate model for aeronautic design, Computers & Fluids, 40 (2011), 195-209. [9] M. Couplet, C. Basdevant and P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modelling, J. Comput. Phys., 207 (2005), 192-220. [10] M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, Cambridge, UK, 2009. [11] E. H. Dowell and K. C. Hall, Modeling of fluid-structure interaction, Annu. Rev. Fluid Mech., 33 (2001), 445-490. [12] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eq., 73 (1988), 309-353. [13] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977. [14] M. A. Grepl and A. T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, ESAIM: M2AN, 39 (2005), 157-181. [15] H. Herrero, Y. Maday and F. Pla, RB (Reduced basis) for RB (Rayleigh-Bénard), Comput. Meth. Appl. Mech. Engrg., 261-262 (2013), 132-141. [16] J. Heyman, G. Girault, Y. Guevel, C. Allery, A. Hamdouni and J. M. Cadou, Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method, Computers & Fluids, 76 (2013), 73-85. [17] P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996. [18] M. Ilak, S. Bagheri, L. Brandt, C. W. Rowley and D. S. Henningson, Model reduction of the nonlinear complex Ginzburg-Landau equation, SIAM J. Appl. Dyn. Sys., 9 (2010), 1284-1302. [19] Y. A. Kutnetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. [20] T. Lieu, C. Farhat and M. Lesoinne, Reduced-order fluid/structure modeling of a complete aircraft configuration, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 5730-5742. [21] D. J. Lucia, P. S. Beran and W. A. Silva, Reduced-order modelling: new approaches for computations physics, Prog. Aerosp. Sci., 40 (2004), 51-117. [22] M. Meis, F. Varas, A. Velazquez and J. M. Vega, Heat transfer enhancement in micro-channels caused by vortex promoters, Int. J. Heat Mass Transfer, 53 (2010), 29-40. [23] M.-L. Rapún and J. M. Vega, Reduced order models based on local POD plus Galerkin projection, J. Comput. Phys., 229 (2010), 3046-3063. [24] M.-L. Rapún, F. Terragni and J. M. Vega, Adaptive POD-based low-dimensional modeling supported by residual estimates, Int. J. Numer. Meth. Engng (2015), in press. [25] D. Rempfer, On low-dimensional Galerkin models for fluid flow, Theor. Comput. Fluid Dyn., 14 (2000), 75-88. [26] S. Sirisup and G. E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models, J. Comput. Phys., 194 (2004), 92-116. [27] S. Sirisup, G. E. Karniadakis, D. Xiu and I. G. Kevrekidis, Equations-free/Galerkin-free POD assisted computation of incompressible flows, J. Comput. Phys., 207 (2005), 568-587. [28] L. Sirovich, Turbulence and the dynamics of coherent structures, Q. Appl. Math., XLV (1987), 561-590. [29] F. Terragni, E. Valero and J. M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, SIAM J. Sci. Comput., 33 (2011), 3538-3561. [30] F. Terragni and J. M. Vega, On the use of POD-based ROMs to analyze bifurcations in some dissipative systems, Physica D, 241 (2012), 1393-1405. [31] F. Terragni and J. M. Vega, Construction of bifurcation diagrams using POD on the fly, SIAM J. Appl. Dyn. Syst., 13 (2014), 339-365.

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##### References:
 [1] E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, Classics Appl. Math. 45, SIAM, Philadelphia, 2003. [2] D. Alonso, J. M. Vega and A. Velazquez, Reduced order model for viscous aerodynamic flow past an airfoil, AIAA J., 48 (2010), 1946-1958. [3] D. Alonso, J. M. Vega, A. Velazquez and V. de Pablo, Reduced-order modeling of three-dimensional external aerodynamic flows, J. Aerospace Engrg., 25 (2012), 588-599. [4] I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 100-142. [5] P. Astrid, S. Weiland, K. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automatic Control, 53 (2008), 2237-2251. [6] M. J. Balajewicz, E. H. Dowell and B. R. Noack, Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation, J. Fluid Mech., 729 (2013), 285-308. [7] M. Bergmann, C. H. Bruneau and A. Iollo, Enablers for robust POD models, J. Comput. Phys., 228 (2009), 516-538. [8] T. Braconnier, M. Ferrier, J. C. Jouhaud, M. Montagnac and P. Sagaut, Towards an adaptive POD/SVD surrogate model for aeronautic design, Computers & Fluids, 40 (2011), 195-209. [9] M. Couplet, C. Basdevant and P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modelling, J. Comput. Phys., 207 (2005), 192-220. [10] M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, Cambridge, UK, 2009. [11] E. H. Dowell and K. C. Hall, Modeling of fluid-structure interaction, Annu. Rev. Fluid Mech., 33 (2001), 445-490. [12] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eq., 73 (1988), 309-353. [13] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977. [14] M. A. Grepl and A. T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, ESAIM: M2AN, 39 (2005), 157-181. [15] H. Herrero, Y. Maday and F. Pla, RB (Reduced basis) for RB (Rayleigh-Bénard), Comput. Meth. Appl. Mech. Engrg., 261-262 (2013), 132-141. [16] J. Heyman, G. Girault, Y. Guevel, C. Allery, A. Hamdouni and J. M. Cadou, Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method, Computers & Fluids, 76 (2013), 73-85. [17] P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996. [18] M. Ilak, S. Bagheri, L. Brandt, C. W. Rowley and D. S. Henningson, Model reduction of the nonlinear complex Ginzburg-Landau equation, SIAM J. Appl. Dyn. Sys., 9 (2010), 1284-1302. [19] Y. A. Kutnetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. [20] T. Lieu, C. Farhat and M. Lesoinne, Reduced-order fluid/structure modeling of a complete aircraft configuration, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 5730-5742. [21] D. J. Lucia, P. S. Beran and W. A. Silva, Reduced-order modelling: new approaches for computations physics, Prog. Aerosp. Sci., 40 (2004), 51-117. [22] M. Meis, F. Varas, A. Velazquez and J. M. Vega, Heat transfer enhancement in micro-channels caused by vortex promoters, Int. J. Heat Mass Transfer, 53 (2010), 29-40. [23] M.-L. Rapún and J. M. Vega, Reduced order models based on local POD plus Galerkin projection, J. Comput. Phys., 229 (2010), 3046-3063. [24] M.-L. Rapún, F. Terragni and J. M. Vega, Adaptive POD-based low-dimensional modeling supported by residual estimates, Int. J. Numer. Meth. Engng (2015), in press. [25] D. Rempfer, On low-dimensional Galerkin models for fluid flow, Theor. Comput. Fluid Dyn., 14 (2000), 75-88. [26] S. Sirisup and G. E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models, J. Comput. Phys., 194 (2004), 92-116. [27] S. Sirisup, G. E. Karniadakis, D. Xiu and I. G. Kevrekidis, Equations-free/Galerkin-free POD assisted computation of incompressible flows, J. Comput. Phys., 207 (2005), 568-587. [28] L. Sirovich, Turbulence and the dynamics of coherent structures, Q. Appl. Math., XLV (1987), 561-590. [29] F. Terragni, E. Valero and J. M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, SIAM J. Sci. Comput., 33 (2011), 3538-3561. [30] F. Terragni and J. M. Vega, On the use of POD-based ROMs to analyze bifurcations in some dissipative systems, Physica D, 241 (2012), 1393-1405. [31] F. Terragni and J. M. Vega, Construction of bifurcation diagrams using POD on the fly, SIAM J. Appl. Dyn. Syst., 13 (2014), 339-365.
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